Journal of Lie Theory

no. 3
A Dixmier-Malliavin Theorem for Lie Groupoids
Journal of Lie Theory, Volume 32 (2022) no. 3, pp. 879-898
A famous theorem of Dixmier-Malliavin asserts that every smooth, compactly-supported function on a Lie group can be expressed as a finite sum in which each term is the convolution, with respect to Haar measure, of two such functions. We establish that the same holds for a Lie groupoid. The analytical heavy lifting is done by a lemma in the original work of Dixmier-Malliavin. We also need the technology of Lie algebroids and the corresponding notion of exponential map. As an application, we obtain a result on the arithmetic of ideals in the smooth convolution algebra of a Lie groupoid arising from functions vanishing to given order on an invariant submanifold of the unit space.
DOI: 10.5802/jolt.1257
Classification: 22A22, 58H05
Keywords: Dixmier-Malliavin, Lie groupoid, convolution 
@article{JOLT_2022_32_3_a13,
     author = {M. D. Francis},
     title = {A {Dixmier-Malliavin} {Theorem} for {Lie} {Groupoids}},
     journal = {Journal of Lie Theory},
     pages = {879--898},
     year = {2022},
     volume = {32},
     number = {3},
     doi = {10.5802/jolt.1257},
     zbl = {1508.58006},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1257/}
}
TY  - JOUR
AU  - M. D. Francis
TI  - A Dixmier-Malliavin Theorem for Lie Groupoids
JO  - Journal of Lie Theory
PY  - 2022
SP  - 879
EP  - 898
VL  - 32
IS  - 3
UR  - https://jolt.centre-mersenne.org/articles/10.5802/jolt.1257/
DO  - 10.5802/jolt.1257
ID  - JOLT_2022_32_3_a13
ER  - 
%0 Journal Article
%A M. D. Francis
%T A Dixmier-Malliavin Theorem for Lie Groupoids
%J Journal of Lie Theory
%D 2022
%P 879-898
%V 32
%N 3
%U https://jolt.centre-mersenne.org/articles/10.5802/jolt.1257/
%R 10.5802/jolt.1257
%F JOLT_2022_32_3_a13
M. D. Francis. A Dixmier-Malliavin Theorem for Lie Groupoids. Journal of Lie Theory, Volume 32 (2022) no. 3, pp. 879-898. doi: 10.5802/jolt.1257

Cited by Sources: